Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system
type: Item
instance of: scholarly article
title: Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system
zbMATH Open document ID: 1203.35030
author: Junjie Wei
author: Junping Shi
author: Fengqi Yi
published in: Journal of Differential Equations
publication date: 2009-03-20T00:00:00Z
review text: The authors investigate a diffusive predator-prey system with Holling type-II functional response and Neumann boundary conditions. For this system the analysis of Hopf bifurcations and steady state bifurcations is carried out in detail, including centre manifold reduction and normal form analysis. It is shown that multiple spatially non-homogeneous period orbits exist and that there exist loops of periodic orbits and steady state solutions. This result can be regarded as evidence for the rich spatio-temporal behaviour observed in that class of systems. The detailed analysis of the bifurcation equations, as it is carried out in this article, can also serve as a valuable resource for learning the applied methods.
reviewed by: Alois Steindl
Mathematics Subject Classification ID: 35B32
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 35B10
Mathematics Subject Classification ID: 92D25
Mathematics Subject Classification ID: 35K51
zbMATH DE Number: 5530626
zbMATH Keywords: Holling type-II functional response
zbMATH Keywords: Hopf bifurcation
zbMATH Keywords: steady state bifurcation
zbMATH Keywords: spatially non-homogeneous periodic orbits
zbMATH Keywords: global bifurcation
zbMATH Keywords: Neumann boundary conditions
MaRDI profile type: Publication
full work available at URL: j.jde.2008.10.024
OpenAlex ID: W2038105662
cites work: Lyapunov functions for a generalized Gause-type model
cites work: Uniqueness of a Limit Cycle for a Predator-Prey System
cites work: Large Time Behavior of Solutions of Systems of Nonlinear Reaction-Diffusion Equations
cites work: Bifurcation from simple eigenvalues
cites work: The Hopf bifurcation theorem in infinite dimensions
cites work: Convergence to Homogeneous Equilibrium State for Generalized Volterra–Lotka Systems with Diffusion
cites work: \(S\)-shaped global bifurcation curve and Hopf bifurcation of positive solutions to a predator-prey model
cites work: Qualitative behaviour of positive solutions of a predator-prey model: Effects of saturation
cites work: Q5474667
cites work: A diffusive predator-prey model with a protection zone
cites work: Allee effect and bistability in a spatially heterogeneous predator-prey model
cites work: Q3895297
cites work: Elliptic partial differential equations
cites work: Q3930015
cites work: Geometric theory of semilinear parabolic equations
cites work: On global stability of a predator-prey system
cites work: A survey of constructing Lyapunov functions for mathematical models in population biology
cites work: Q5471369
cites work: Competing Predators
cites work: Bifurcation theory. An introduction with applications of PDEs
cites work: Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a prey refuge
cites work: Uniqueness of limit cycles in Gause-type models of predator-prey systems
cites work: Large amplitude stationary solutions to a chemotaxis system
cites work: Imperfect transcritical and pitchfork bifurcations
cites work: Diffusion, self-diffusion and cross-diffusion
cites work: Q1233671
cites work: Spatiotemporal Complexity of Plankton and Fish Dynamics
cites work: Degree theory for \(C^1\) Fredholm mappings of index 0
cites work: Stationary Pattern of a Ratio-Dependent Food Chain Model with Diffusion
cites work: On stationary patterns of a reaction–diffusion model with autocatalysis and saturation law
cites work: Some global results for nonlinear eigenvalue problems
cites work: On global bifurcation for quasilinear elliptic systems on bounded domains
cites work: The chemical basis of morphogenesis
cites work: Symmetric functional differential equations and neural networks with memory
cites work: Diffusion-driven instability and bifurcation in the Lengyel-Epstein system
cites work: Global asymptotical behavior of the Lengyel-Epstein reaction-diffusion system
cites work: Proof of the uniqueness theorem of limit cycles of generalized liénard equations
DOI: 10.1016/J.JDE.2008.10.024
description: scientific article; zbMATH DE number 5530626